Computer simulation of fluid flow in porous media is widely used in the oil industry, in hydrology, and in environmental studies for remediation of contaminated groundwater. Simulation predictions often have a significant impact on the economic valuation of assets, plans for depletion of hydrocarbon assets and government policies.
Hydrocarbon deposits, such as oil and gas, are found in nature in complex underground structures known as “reservoirs.” Reservoirs are comprised of various types of porous media (rocks) with different physical properties, such as porosity and permeability. These properties may vary widely over short distances. Fluid flow in a reservoir is determined by the physical properties.
The development of stochastic geologic property modeling techniques by geologists has allowed modelers to create subsurface models with a tremendous amount of data, which is represented in a three-dimensional “grid” that overlays the subsurface volume. It is not practical to perform reservoir simulations for the various situations of interest at geologic model scale, because of the large number of cells in the geologic model grid. Also, complex property distributions have made simple permeability averaging techniques obsolete. Therefore, “upscaling” (the formation of coarser grids for flow calculations) has become an integral part of reservoir simulation.
Assembling data describing rock properties and geologic structures is a crucial step toward accurate simulations of fluid flow in reservoirs. The geo-cellular models that assemble the data include rock properties (e.g., porosity and permeability) defined in each cell. The geologic cells form a non-overlapping partition of a reservoir.
The geo-cellular model may include millions of geologic cells to describe a reservoir, so direct simulation of reservoir fluid movement for the many cases of interest is cost-prohibitive. Thus, from an economic standpoint it is necessary to transform a detailed geologic model into a coarse simulation model with fewer degrees of freedom, so that reservoir simulation can be performed at an acceptable cost. This transformation is called both “scaleup” and “upscaling.” Recent reviews of scaleup have been published by D. Stern (“Practical Aspects of Scaleup of Simulation Models,” J. Pet. Tech., September 2005, pp. 74-82) and L. J. Durlofsky (“Upscaling and Gridding of Fine Scale Geologic Models for Flow Simulation,” paper presented at 8th Int'l Forum on Reservoir Simulation, Stressa, Italy, June, 2005) (See: http://ekofisk.stanford.edu/faculty/durlofskypub12.html).
Upscaling involves building a simulation grid that is coarser than the geologic grid and converting properties defined on the geologic grid to the simulation grid. Once a simulation grid is defined, converting geologic properties typically requires that certain averages of the geologic properties be calculated to populate the simulation grid. For some of the properties, such as porosity, simple averages with suitable weights are sufficient. To scaleup permeability, flow-based averaging procedures have proven to be the best way. Durlofsky (2005) reviews such procedures and a recent mathematical analysis of flow-based permeability-scaleup is given by Wu et al. (“Analysis of Upscaling Absolute Permeability,” Discrete and Continuous Dynamical Systems-Series B, Vol. 2, No. 2, 2002).
Flow-based scaleup requires solving single-phase Darcy flow equations on a fine-scale grid. Most of the existing methods require the fine grid to be aligned with the coarse simulation grid. Recently, a method of upscaling simulation grid transmissibility using flow solutions defined on a fine grid that is not aligned with the simulation grid was described by He (C. He, “Structured Flow-based Gridding and Upscaling for Reservoir Simulation,” PhD Thesis, Stanford University, Stanford Calif., December, 2004). White and Horne present an algorithm to compute scaled-up values of transmissibility when there is permeability heterogeneity and anisotropy at the fine-grid scale (“Computing Absolute Transmissibility in the Presence of Fine-Scale Heterogeneity,” paper SPE 16011, Ninth SPE Symposium on Reservoir Simulation, Society of Petroleum Engineers, 209-220 (1987)).
As discussed by Stern (2005) and Durlofsky (2005), a successful scaleup often requires a simulation grid that is capable of capturing correlated heterogeneities directly. An iterative procedure is often required, which involves building multiple simulation grids to determine the “optimum” grid. This process is called grid optimization. Building multiple simulation grids requires repeated scaleup of the geologic model. For permeability scaleup, generating flow solutions on a fine-scale grid is the most time-consuming and costly step. Due to its high cost, automatic grid optimization is not feasible; in fact, even manual changes of simulation grids are seldom done in practice. As a result, simulation models often do not have the best accuracy, and they may produce predictions that are not consistent with the geologic models. What is needed is a method that allows faster and lower cost grid optimization.